Question: Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{y^2 - 3y - 28}{y^2 + 13y + 36}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{y^2 - 3y - 28}{y^2 + 13y + 36} = \dfrac{(y - 7)(y + 4)}{(y + 9)(y + 4)} $ Notice that the term $(y + 4)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(y + 4)$ gives: $r = \dfrac{y - 7}{y + 9}$ Since we divided by $(y + 4)$, $y \neq -4$. $r = \dfrac{y - 7}{y + 9}; \space y \neq -4$